LAPACK  3.9.0
LAPACK: Linear Algebra PACKage
dsygvx.f
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1 *> \brief \b DSYGVX
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DSYGVX + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygvx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
22 * VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
23 * LWORK, IWORK, IFAIL, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER JOBZ, RANGE, UPLO
27 * INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
28 * DOUBLE PRECISION ABSTOL, VL, VU
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IFAIL( * ), IWORK( * )
32 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
33 * $ Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> DSYGVX computes selected eigenvalues, and optionally, eigenvectors
43 *> of a real generalized symmetric-definite eigenproblem, of the form
44 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
45 *> and B are assumed to be symmetric and B is also positive definite.
46 *> Eigenvalues and eigenvectors can be selected by specifying either a
47 *> range of values or a range of indices for the desired eigenvalues.
48 *> \endverbatim
49 *
50 * Arguments:
51 * ==========
52 *
53 *> \param[in] ITYPE
54 *> \verbatim
55 *> ITYPE is INTEGER
56 *> Specifies the problem type to be solved:
57 *> = 1: A*x = (lambda)*B*x
58 *> = 2: A*B*x = (lambda)*x
59 *> = 3: B*A*x = (lambda)*x
60 *> \endverbatim
61 *>
62 *> \param[in] JOBZ
63 *> \verbatim
64 *> JOBZ is CHARACTER*1
65 *> = 'N': Compute eigenvalues only;
66 *> = 'V': Compute eigenvalues and eigenvectors.
67 *> \endverbatim
68 *>
69 *> \param[in] RANGE
70 *> \verbatim
71 *> RANGE is CHARACTER*1
72 *> = 'A': all eigenvalues will be found.
73 *> = 'V': all eigenvalues in the half-open interval (VL,VU]
74 *> will be found.
75 *> = 'I': the IL-th through IU-th eigenvalues will be found.
76 *> \endverbatim
77 *>
78 *> \param[in] UPLO
79 *> \verbatim
80 *> UPLO is CHARACTER*1
81 *> = 'U': Upper triangle of A and B are stored;
82 *> = 'L': Lower triangle of A and B are stored.
83 *> \endverbatim
84 *>
85 *> \param[in] N
86 *> \verbatim
87 *> N is INTEGER
88 *> The order of the matrix pencil (A,B). N >= 0.
89 *> \endverbatim
90 *>
91 *> \param[in,out] A
92 *> \verbatim
93 *> A is DOUBLE PRECISION array, dimension (LDA, N)
94 *> On entry, the symmetric matrix A. If UPLO = 'U', the
95 *> leading N-by-N upper triangular part of A contains the
96 *> upper triangular part of the matrix A. If UPLO = 'L',
97 *> the leading N-by-N lower triangular part of A contains
98 *> the lower triangular part of the matrix A.
99 *>
100 *> On exit, the lower triangle (if UPLO='L') or the upper
101 *> triangle (if UPLO='U') of A, including the diagonal, is
102 *> destroyed.
103 *> \endverbatim
104 *>
105 *> \param[in] LDA
106 *> \verbatim
107 *> LDA is INTEGER
108 *> The leading dimension of the array A. LDA >= max(1,N).
109 *> \endverbatim
110 *>
111 *> \param[in,out] B
112 *> \verbatim
113 *> B is DOUBLE PRECISION array, dimension (LDB, N)
114 *> On entry, the symmetric matrix B. If UPLO = 'U', the
115 *> leading N-by-N upper triangular part of B contains the
116 *> upper triangular part of the matrix B. If UPLO = 'L',
117 *> the leading N-by-N lower triangular part of B contains
118 *> the lower triangular part of the matrix B.
119 *>
120 *> On exit, if INFO <= N, the part of B containing the matrix is
121 *> overwritten by the triangular factor U or L from the Cholesky
122 *> factorization B = U**T*U or B = L*L**T.
123 *> \endverbatim
124 *>
125 *> \param[in] LDB
126 *> \verbatim
127 *> LDB is INTEGER
128 *> The leading dimension of the array B. LDB >= max(1,N).
129 *> \endverbatim
130 *>
131 *> \param[in] VL
132 *> \verbatim
133 *> VL is DOUBLE PRECISION
134 *> If RANGE='V', the lower bound of the interval to
135 *> be searched for eigenvalues. VL < VU.
136 *> Not referenced if RANGE = 'A' or 'I'.
137 *> \endverbatim
138 *>
139 *> \param[in] VU
140 *> \verbatim
141 *> VU is DOUBLE PRECISION
142 *> If RANGE='V', the upper bound of the interval to
143 *> be searched for eigenvalues. VL < VU.
144 *> Not referenced if RANGE = 'A' or 'I'.
145 *> \endverbatim
146 *>
147 *> \param[in] IL
148 *> \verbatim
149 *> IL is INTEGER
150 *> If RANGE='I', the index of the
151 *> smallest eigenvalue to be returned.
152 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
153 *> Not referenced if RANGE = 'A' or 'V'.
154 *> \endverbatim
155 *>
156 *> \param[in] IU
157 *> \verbatim
158 *> IU is INTEGER
159 *> If RANGE='I', the index of the
160 *> largest eigenvalue to be returned.
161 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
162 *> Not referenced if RANGE = 'A' or 'V'.
163 *> \endverbatim
164 *>
165 *> \param[in] ABSTOL
166 *> \verbatim
167 *> ABSTOL is DOUBLE PRECISION
168 *> The absolute error tolerance for the eigenvalues.
169 *> An approximate eigenvalue is accepted as converged
170 *> when it is determined to lie in an interval [a,b]
171 *> of width less than or equal to
172 *>
173 *> ABSTOL + EPS * max( |a|,|b| ) ,
174 *>
175 *> where EPS is the machine precision. If ABSTOL is less than
176 *> or equal to zero, then EPS*|T| will be used in its place,
177 *> where |T| is the 1-norm of the tridiagonal matrix obtained
178 *> by reducing C to tridiagonal form, where C is the symmetric
179 *> matrix of the standard symmetric problem to which the
180 *> generalized problem is transformed.
181 *>
182 *> Eigenvalues will be computed most accurately when ABSTOL is
183 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
184 *> If this routine returns with INFO>0, indicating that some
185 *> eigenvectors did not converge, try setting ABSTOL to
186 *> 2*DLAMCH('S').
187 *> \endverbatim
188 *>
189 *> \param[out] M
190 *> \verbatim
191 *> M is INTEGER
192 *> The total number of eigenvalues found. 0 <= M <= N.
193 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
194 *> \endverbatim
195 *>
196 *> \param[out] W
197 *> \verbatim
198 *> W is DOUBLE PRECISION array, dimension (N)
199 *> On normal exit, the first M elements contain the selected
200 *> eigenvalues in ascending order.
201 *> \endverbatim
202 *>
203 *> \param[out] Z
204 *> \verbatim
205 *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
206 *> If JOBZ = 'N', then Z is not referenced.
207 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
208 *> contain the orthonormal eigenvectors of the matrix A
209 *> corresponding to the selected eigenvalues, with the i-th
210 *> column of Z holding the eigenvector associated with W(i).
211 *> The eigenvectors are normalized as follows:
212 *> if ITYPE = 1 or 2, Z**T*B*Z = I;
213 *> if ITYPE = 3, Z**T*inv(B)*Z = I.
214 *>
215 *> If an eigenvector fails to converge, then that column of Z
216 *> contains the latest approximation to the eigenvector, and the
217 *> index of the eigenvector is returned in IFAIL.
218 *> Note: the user must ensure that at least max(1,M) columns are
219 *> supplied in the array Z; if RANGE = 'V', the exact value of M
220 *> is not known in advance and an upper bound must be used.
221 *> \endverbatim
222 *>
223 *> \param[in] LDZ
224 *> \verbatim
225 *> LDZ is INTEGER
226 *> The leading dimension of the array Z. LDZ >= 1, and if
227 *> JOBZ = 'V', LDZ >= max(1,N).
228 *> \endverbatim
229 *>
230 *> \param[out] WORK
231 *> \verbatim
232 *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
233 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
234 *> \endverbatim
235 *>
236 *> \param[in] LWORK
237 *> \verbatim
238 *> LWORK is INTEGER
239 *> The length of the array WORK. LWORK >= max(1,8*N).
240 *> For optimal efficiency, LWORK >= (NB+3)*N,
241 *> where NB is the blocksize for DSYTRD returned by ILAENV.
242 *>
243 *> If LWORK = -1, then a workspace query is assumed; the routine
244 *> only calculates the optimal size of the WORK array, returns
245 *> this value as the first entry of the WORK array, and no error
246 *> message related to LWORK is issued by XERBLA.
247 *> \endverbatim
248 *>
249 *> \param[out] IWORK
250 *> \verbatim
251 *> IWORK is INTEGER array, dimension (5*N)
252 *> \endverbatim
253 *>
254 *> \param[out] IFAIL
255 *> \verbatim
256 *> IFAIL is INTEGER array, dimension (N)
257 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
258 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
259 *> indices of the eigenvectors that failed to converge.
260 *> If JOBZ = 'N', then IFAIL is not referenced.
261 *> \endverbatim
262 *>
263 *> \param[out] INFO
264 *> \verbatim
265 *> INFO is INTEGER
266 *> = 0: successful exit
267 *> < 0: if INFO = -i, the i-th argument had an illegal value
268 *> > 0: DPOTRF or DSYEVX returned an error code:
269 *> <= N: if INFO = i, DSYEVX failed to converge;
270 *> i eigenvectors failed to converge. Their indices
271 *> are stored in array IFAIL.
272 *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
273 *> minor of order i of B is not positive definite.
274 *> The factorization of B could not be completed and
275 *> no eigenvalues or eigenvectors were computed.
276 *> \endverbatim
277 *
278 * Authors:
279 * ========
280 *
281 *> \author Univ. of Tennessee
282 *> \author Univ. of California Berkeley
283 *> \author Univ. of Colorado Denver
284 *> \author NAG Ltd.
285 *
286 *> \date June 2016
287 *
288 *> \ingroup doubleSYeigen
289 *
290 *> \par Contributors:
291 * ==================
292 *>
293 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
294 *
295 * =====================================================================
296  SUBROUTINE dsygvx( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
297  $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
298  $ LWORK, IWORK, IFAIL, INFO )
299 *
300 * -- LAPACK driver routine (version 3.7.0) --
301 * -- LAPACK is a software package provided by Univ. of Tennessee, --
302 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
303 * June 2016
304 *
305 * .. Scalar Arguments ..
306  CHARACTER JOBZ, RANGE, UPLO
307  INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
308  DOUBLE PRECISION ABSTOL, VL, VU
309 * ..
310 * .. Array Arguments ..
311  INTEGER IFAIL( * ), IWORK( * )
312  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
313  $ z( ldz, * )
314 * ..
315 *
316 * =====================================================================
317 *
318 * .. Parameters ..
319  DOUBLE PRECISION ONE
320  PARAMETER ( ONE = 1.0d+0 )
321 * ..
322 * .. Local Scalars ..
323  LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
324  CHARACTER TRANS
325  INTEGER LWKMIN, LWKOPT, NB
326 * ..
327 * .. External Functions ..
328  LOGICAL LSAME
329  INTEGER ILAENV
330  EXTERNAL lsame, ilaenv
331 * ..
332 * .. External Subroutines ..
333  EXTERNAL dpotrf, dsyevx, dsygst, dtrmm, dtrsm, xerbla
334 * ..
335 * .. Intrinsic Functions ..
336  INTRINSIC max, min
337 * ..
338 * .. Executable Statements ..
339 *
340 * Test the input parameters.
341 *
342  upper = lsame( uplo, 'U' )
343  wantz = lsame( jobz, 'V' )
344  alleig = lsame( range, 'A' )
345  valeig = lsame( range, 'V' )
346  indeig = lsame( range, 'I' )
347  lquery = ( lwork.EQ.-1 )
348 *
349  info = 0
350  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
351  info = -1
352  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
353  info = -2
354  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
355  info = -3
356  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
357  info = -4
358  ELSE IF( n.LT.0 ) THEN
359  info = -5
360  ELSE IF( lda.LT.max( 1, n ) ) THEN
361  info = -7
362  ELSE IF( ldb.LT.max( 1, n ) ) THEN
363  info = -9
364  ELSE
365  IF( valeig ) THEN
366  IF( n.GT.0 .AND. vu.LE.vl )
367  $ info = -11
368  ELSE IF( indeig ) THEN
369  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
370  info = -12
371  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
372  info = -13
373  END IF
374  END IF
375  END IF
376  IF (info.EQ.0) THEN
377  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
378  info = -18
379  END IF
380  END IF
381 *
382  IF( info.EQ.0 ) THEN
383  lwkmin = max( 1, 8*n )
384  nb = ilaenv( 1, 'DSYTRD', uplo, n, -1, -1, -1 )
385  lwkopt = max( lwkmin, ( nb + 3 )*n )
386  work( 1 ) = lwkopt
387 *
388  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
389  info = -20
390  END IF
391  END IF
392 *
393  IF( info.NE.0 ) THEN
394  CALL xerbla( 'DSYGVX', -info )
395  RETURN
396  ELSE IF( lquery ) THEN
397  RETURN
398  END IF
399 *
400 * Quick return if possible
401 *
402  m = 0
403  IF( n.EQ.0 ) THEN
404  RETURN
405  END IF
406 *
407 * Form a Cholesky factorization of B.
408 *
409  CALL dpotrf( uplo, n, b, ldb, info )
410  IF( info.NE.0 ) THEN
411  info = n + info
412  RETURN
413  END IF
414 *
415 * Transform problem to standard eigenvalue problem and solve.
416 *
417  CALL dsygst( itype, uplo, n, a, lda, b, ldb, info )
418  CALL dsyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
419  $ m, w, z, ldz, work, lwork, iwork, ifail, info )
420 *
421  IF( wantz ) THEN
422 *
423 * Backtransform eigenvectors to the original problem.
424 *
425  IF( info.GT.0 )
426  $ m = info - 1
427  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
428 *
429 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
430 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
431 *
432  IF( upper ) THEN
433  trans = 'N'
434  ELSE
435  trans = 'T'
436  END IF
437 *
438  CALL dtrsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
439  $ ldb, z, ldz )
440 *
441  ELSE IF( itype.EQ.3 ) THEN
442 *
443 * For B*A*x=(lambda)*x;
444 * backtransform eigenvectors: x = L*y or U**T*y
445 *
446  IF( upper ) THEN
447  trans = 'T'
448  ELSE
449  trans = 'N'
450  END IF
451 *
452  CALL dtrmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
453  $ ldb, z, ldz )
454  END IF
455  END IF
456 *
457 * Set WORK(1) to optimal workspace size.
458 *
459  work( 1 ) = lwkopt
460 *
461  RETURN
462 *
463 * End of DSYGVX
464 *
465  END
dtrsm
subroutine dtrsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRSM
Definition: dtrsm.f:183
dtrmm
subroutine dtrmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
DTRMM
Definition: dtrmm.f:179
dsygst
subroutine dsygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
DSYGST
Definition: dsygst.f:129
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
dsygvx
subroutine dsygvx(ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYGVX
Definition: dsygvx.f:299
dpotrf
subroutine dpotrf(UPLO, N, A, LDA, INFO)
DPOTRF
Definition: dpotrf.f:109
dsyevx
subroutine dsyevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
Definition: dsyevx.f:255